McMillan's analysis.
McMillan in 1968 provided a seminal analysis[1]} of T\(_c\)  and its dependence on \(\alpha ^2F(\omega )\) , presenting his iconic “McMillan equation” expressing T\(_c\)  via two materials quantities, the phonon frequency \(\theta \)  and \(\lambda \) . A phenomenological retarded Coulomb repulsion \(\mu ^*\)  was included in the Eliashberg equation that was solved to relate T\(_c\)  to \(\alpha ^2F\) . The results for various \(\alpha ^2F\)  functions were fit to a generalization of the BCS equation of the form
\(T_c^{McM} &=& \frac{\theta }{1.45}exp\left[ -1.04\frac{1+\lambda }{\lambda -\mu ^* - 0.62\lambda \mu ^*}\right] \nonumber \\&\equiv & \frac{\theta }{1.45}exp \left(-\frac{1+\lambda }{\lambda -\mu _{eff}^*}\right)^{1.04}\) 
